Optimal. Leaf size=65 \[ -\sqrt{x^2-x-1}-\tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right ) \]
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Rubi [A] time = 0.0428585, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {734, 843, 621, 206, 724, 204} \[ -\sqrt{x^2-x-1}-\tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 734
Rule 843
Rule 621
Rule 206
Rule 724
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{-1-x+x^2}}{1-x} \, dx &=-\sqrt{-1-x+x^2}+\frac{1}{2} \int \frac{-3+x}{(1-x) \sqrt{-1-x+x^2}} \, dx\\ &=-\sqrt{-1-x+x^2}-\frac{1}{2} \int \frac{1}{\sqrt{-1-x+x^2}} \, dx-\int \frac{1}{(1-x) \sqrt{-1-x+x^2}} \, dx\\ &=-\sqrt{-1-x+x^2}+2 \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,\frac{3-x}{\sqrt{-1-x+x^2}}\right )-\operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{-1+2 x}{\sqrt{-1-x+x^2}}\right )\\ &=-\sqrt{-1-x+x^2}-\tan ^{-1}\left (\frac{3-x}{2 \sqrt{-1-x+x^2}}\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{-1+2 x}{2 \sqrt{-1-x+x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0137811, size = 65, normalized size = 1. \[ -\sqrt{x^2-x-1}-\tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{2 x-1}{2 \sqrt{x^2-x-1}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 46, normalized size = 0.7 \begin{align*} -\sqrt{ \left ( -1+x \right ) ^{2}-2+x}-{\frac{1}{2}\ln \left ( -{\frac{1}{2}}+x+\sqrt{ \left ( -1+x \right ) ^{2}-2+x} \right ) }+\arctan \left ({\frac{-3+x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}-2+x}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51435, size = 78, normalized size = 1.2 \begin{align*} -\sqrt{x^{2} - x - 1} + \arcsin \left (\frac{\sqrt{5} x}{5 \,{\left | x - 1 \right |}} - \frac{3 \, \sqrt{5}}{5 \,{\left | x - 1 \right |}}\right ) - \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - x - 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48509, size = 136, normalized size = 2.09 \begin{align*} -\sqrt{x^{2} - x - 1} + 2 \, \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) + \frac{1}{2} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sqrt{x^{2} - x - 1}}{x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10441, size = 70, normalized size = 1.08 \begin{align*} -\sqrt{x^{2} - x - 1} + 2 \, \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) + \frac{1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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